Question about Uncle Tom’s Cabin

So I’ve reached the part in the story where (I guess I should say spoilers ahead of you haven’t read it) Cassy faints on the boat as she realizes Eliza is her daughter.

I’m really confused about this for a few reasons. First, when Cassy told her story to Tom, she only mentioned having a daughter named “Elise,” and also, she realizes Eliza is her daughter only after George mentions a person named “Simmons.”

Are we to assume that “Elise” is merely a nickname? And additionally, who is “Simmons?” I don’t ever remember hearing of a character named that.

Hopefully can clear this up for me? I just like to have everything squared away in terms of understanding what I’m supposed to. Thanks!